Equivalence of Definitions of Top of Lattice
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Theorem
The following definitions of the concept of Top of Lattice are equivalent:
Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.
Definition 1
Let $S$ admit a greatest element $\top$.
Then $\top$ is called the top of $S$.
Definition 2
Let $\wedge$ have an identity element $\top$.
Then $\top$ is called the top of $S$.
Proof
By definition, $\top$ is the greatest element of $S$ if and only if for all $a \in S$:
- $a \preceq \top$
By Ordering in terms of Meet, this is equivalent to:
- $a \wedge \top = a$
If this equality holds for all $a \in S$, then by definition $\top$ is an identity for $\wedge$.
The result follows.
$\blacksquare$